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Mirrors > Home > ILE Home > Th. List > brcogw | Unicode version |
Description: Ordered pair membership in a composition. (Contributed by Thierry Arnoux, 14-Jan-2018.) |
Ref | Expression |
---|---|
brcogw |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl1 1001 |
. 2
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2 | simpl2 1002 |
. 2
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3 | breq2 4019 |
. . . . . 6
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4 | breq1 4018 |
. . . . . 6
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5 | 3, 4 | anbi12d 473 |
. . . . 5
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6 | 5 | spcegv 2837 |
. . . 4
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7 | 6 | imp 124 |
. . 3
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8 | 7 | 3ad2antl3 1162 |
. 2
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9 | brcog 4806 |
. . 3
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10 | 9 | biimpar 297 |
. 2
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11 | 1, 2, 8, 10 | syl21anc 1247 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-v 2751 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-br 4016 df-opab 4077 df-co 4647 |
This theorem is referenced by: (None) |
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